Graph Theory: Euler Circuits

1. Graph Theory: Euler Circuits Christina Mende Math 480 April 15, 2013

2. Preliminaries a V={a,b,c,d} E={{a,b},{a,c},{b,c},{c,d}} d b c • A graph is a pair G=(V,E) of sets satisfying • The elements of V are the vertices of our graph. • The elements of E are the edges of our graph.

3. Preliminaries deg(a)=2 deg(b)=2 deg(c)=3 deg(d)=1 a V={a,b,c,d} E={{a,b},{a,c},{b,c},{c,d}} d b c The degree of a vertex v (denoted deg(v))is the number of edges directly connected to that vertex.

4. Preliminaries a e.g. a,b,c,d is a path a,b,c,a is a circuit. d b c A path of length n>0 is a sequence of edges that begins at a vertex of the graph and travels from vertex to vertex along the edges of a graph. If a path begins and ends at the same vertex, it is called a closed path or circuit.

5. Preliminaries G1 G2 a Disconnected Connected In other words, a graph is connected if there is a path between every distinct vertex of the graph. d b c A graph is connected if it cannot be expressed as the union of two graphs

6. The Seven Bridges of Königsberg C D A Multigraph model of Seven Bridges of Konigsberg. B Question: Is it possible to start at some location in town, travel across all seven bridges without crossing any bridge twice, and return to the same starting point?

7. Euler Circuits A connected graph G is called Eulerian if there exists a closed path which includes every edge of G. Note that this means each edge must be traversed once and only. Such a path is called an Eulerian circuit.

8. Euler Circuits C D A B Theorem: A connected graph G is Eulerian if and only if the degree of every vertex of G is even.

9. Euler Paths Constructions:Fleury’s Algorithm Note: a bridge is an edge such that, if you cross it, the remaining graph will be disconnected. • If G is an Eulerian graph, then we can always construct an Euler circuit for G using the following algorithm. • Pick any vertex v to start and traverse the edges in an arbitrary manner, subject to the following rules: • Erase the edges as they are traversed, • Never cross a bridge of the reduced graph unless there is no other choice, • Avoid edges that will prematurely end the circuit.

10. Fleury’s Algorithm • Pick any vertex v to start and traverse the edges in an arbitrary manner, subject to the following rules: • Erase the edges as they are traversed, • Never cross a bridge of the reduced graph unless there is no other choice, • Avoid edges that will prematurely end the circuit.

11. Hierholzer’s Algorithm Circuit 1: A , F , D , C , A A B Circuit 2: , G , E , A , B , F , D F , G , B Completed Euler Circuit: A, C, F, G, B, D, G, E, B, A, F, D, A D E C G F • Pick any vertex v to start • From v, pick an edge to traverse, (remember to darken those edges), and continue until you return to the starting vertex v. • If you have traveled through all the edges, you’re done! • If not, pick any vertex from the the darkened graph that still has edges not traveled and repeat step 2 using this new vertex. • Splice together the two circuits. If you have traveled all the edges of the graph, you’re done. If not, repeat the process.

12. Any Questions? Image from: http://www.cs.sunysb.edu/~skiena/combinatorica/animations/euler.html • References • R. Diestel, Graph Theory. • L.R. Foulds, Graph Theory Applications. • K. Rosen, Discrete Mathematics and Its Applications. • R. Wilson, Introduction To Graph Theory.